3.7.30 \(\int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^6} \, dx\)

Optimal. Leaf size=283 \[ -\frac {(7 a d+3 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{9/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (7 a d+3 b c) (b c-a d)^3}{128 a^2 c^4 x}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)^2}{64 a c^4 x^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 a c^3 x^3}+\frac {(a+b x)^{5/2} (c+d x)^{3/2} (7 a d+3 b c)}{40 a c^2 x^4}-\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5} \]

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Rubi [A]  time = 0.15, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} (7 a d+3 b c) (b c-a d)^3}{128 a^2 c^4 x}-\frac {(7 a d+3 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{9/2}}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)^2}{64 a c^4 x^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 a c^3 x^3}+\frac {(a+b x)^{5/2} (c+d x)^{3/2} (7 a d+3 b c)}{40 a c^2 x^4}-\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*Sqrt[c + d*x])/x^6,x]

[Out]

((b*c - a*d)^3*(3*b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^2*c^4*x) + ((b*c - a*d)^2*(3*b*c + 7*a*d)*S
qrt[a + b*x]*(c + d*x)^(3/2))/(64*a*c^4*x^2) + ((b*c - a*d)*(3*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(
48*a*c^3*x^3) + ((3*b*c + 7*a*d)*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(40*a*c^2*x^4) - ((a + b*x)^(7/2)*(c + d*x)^
(3/2))/(5*a*c*x^5) - ((b*c - a*d)^4*(3*b*c + 7*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/
(128*a^(5/2)*c^(9/2))

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^6} \, dx &=-\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5}-\frac {\left (\frac {3 b c}{2}+\frac {7 a d}{2}\right ) \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^5} \, dx}{5 a c}\\ &=\frac {(3 b c+7 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 a c^2 x^4}-\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5}-\frac {((b c-a d) (3 b c+7 a d)) \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^4} \, dx}{16 a c^2}\\ &=\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 a c^3 x^3}+\frac {(3 b c+7 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 a c^2 x^4}-\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5}-\frac {\left ((b c-a d)^2 (3 b c+7 a d)\right ) \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3} \, dx}{32 a c^3}\\ &=\frac {(b c-a d)^2 (3 b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^4 x^2}+\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 a c^3 x^3}+\frac {(3 b c+7 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 a c^2 x^4}-\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5}-\frac {\left ((b c-a d)^3 (3 b c+7 a d)\right ) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{128 a c^4}\\ &=\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^4 x}+\frac {(b c-a d)^2 (3 b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^4 x^2}+\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 a c^3 x^3}+\frac {(3 b c+7 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 a c^2 x^4}-\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5}+\frac {\left ((b c-a d)^4 (3 b c+7 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^2 c^4}\\ &=\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^4 x}+\frac {(b c-a d)^2 (3 b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^4 x^2}+\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 a c^3 x^3}+\frac {(3 b c+7 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 a c^2 x^4}-\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5}+\frac {\left ((b c-a d)^4 (3 b c+7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 a^2 c^4}\\ &=\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^2 c^4 x}+\frac {(b c-a d)^2 (3 b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a c^4 x^2}+\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 a c^3 x^3}+\frac {(3 b c+7 a d) (a+b x)^{5/2} (c+d x)^{3/2}}{40 a c^2 x^4}-\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5}-\frac {(b c-a d)^4 (3 b c+7 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{5/2} c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 215, normalized size = 0.76 \begin {gather*} -\frac {\frac {(7 a d+3 b c) \left (5 x (b c-a d) \left (\frac {3 x (b c-a d) \left (x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} (2 a c+a d x+b c x)\right )}{a^{3/2} c^{3/2}}-8 (a+b x)^{3/2} (c+d x)^{3/2}\right )-48 c (a+b x)^{5/2} (c+d x)^{3/2}\right )}{384 c^2 x^4}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{x^5}}{5 a c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*Sqrt[c + d*x])/x^6,x]

[Out]

-1/5*(((a + b*x)^(7/2)*(c + d*x)^(3/2))/x^5 + ((3*b*c + 7*a*d)*(-48*c*(a + b*x)^(5/2)*(c + d*x)^(3/2) + 5*(b*c
- a*d)*x*(-8*(a + b*x)^(3/2)*(c + d*x)^(3/2) + (3*(b*c - a*d)*x*(-(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x
]*(2*a*c + b*c*x + a*d*x)) + (b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(
3/2)*c^(3/2)))))/(384*c^2*x^4))/(a*c)

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IntegrateAlgebraic [A]  time = 0.45, size = 307, normalized size = 1.08 \begin {gather*} \frac {\sqrt {c+d x} (a d-b c)^4 \left (\frac {105 a^5 d (c+d x)^4}{(a+b x)^4}+\frac {45 a^4 b c (c+d x)^4}{(a+b x)^4}-\frac {490 a^4 c d (c+d x)^3}{(a+b x)^3}-\frac {210 a^3 b c^2 (c+d x)^3}{(a+b x)^3}+\frac {896 a^3 c^2 d (c+d x)^2}{(a+b x)^2}+\frac {384 a^2 b c^3 (c+d x)^2}{(a+b x)^2}-\frac {790 a^2 c^3 d (c+d x)}{a+b x}+\frac {210 a b c^4 (c+d x)}{a+b x}-105 a c^4 d-45 b c^5\right )}{1920 a^2 c^4 \sqrt {a+b x} \left (\frac {a (c+d x)}{a+b x}-c\right )^5}-\frac {(a d-b c)^4 (7 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{5/2} c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(5/2)*Sqrt[c + d*x])/x^6,x]

[Out]

((-(b*c) + a*d)^4*Sqrt[c + d*x]*(-45*b*c^5 - 105*a*c^4*d + (210*a*b*c^4*(c + d*x))/(a + b*x) - (790*a^2*c^3*d*
(c + d*x))/(a + b*x) + (384*a^2*b*c^3*(c + d*x)^2)/(a + b*x)^2 + (896*a^3*c^2*d*(c + d*x)^2)/(a + b*x)^2 - (21
0*a^3*b*c^2*(c + d*x)^3)/(a + b*x)^3 - (490*a^4*c*d*(c + d*x)^3)/(a + b*x)^3 + (45*a^4*b*c*(c + d*x)^4)/(a + b
*x)^4 + (105*a^5*d*(c + d*x)^4)/(a + b*x)^4))/(1920*a^2*c^4*Sqrt[a + b*x]*(-c + (a*(c + d*x))/(a + b*x))^5) -
((-(b*c) + a*d)^4*(3*b*c + 7*a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(128*a^(5/2)*c^(9/
2))

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fricas [A]  time = 19.05, size = 732, normalized size = 2.59 \begin {gather*} \left [\frac {15 \, {\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {a c} x^{5} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (384 \, a^{5} c^{5} - {\left (45 \, a b^{4} c^{5} - 60 \, a^{2} b^{3} c^{4} d + 346 \, a^{3} b^{2} c^{3} d^{2} - 340 \, a^{4} b c^{2} d^{3} + 105 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (15 \, a^{2} b^{3} c^{5} + 109 \, a^{3} b^{2} c^{4} d - 111 \, a^{4} b c^{3} d^{2} + 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{2} c^{5} + 22 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (21 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{3} c^{5} x^{5}}, \frac {15 \, {\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (384 \, a^{5} c^{5} - {\left (45 \, a b^{4} c^{5} - 60 \, a^{2} b^{3} c^{4} d + 346 \, a^{3} b^{2} c^{3} d^{2} - 340 \, a^{4} b c^{2} d^{3} + 105 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (15 \, a^{2} b^{3} c^{5} + 109 \, a^{3} b^{2} c^{4} d - 111 \, a^{4} b c^{3} d^{2} + 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{2} c^{5} + 22 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (21 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{3} c^{5} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(1/2)/x^6,x, algorithm="fricas")

[Out]

[1/7680*(15*(3*b^5*c^5 - 5*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 30*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 7*a^5*d^5)
*sqrt(a*c)*x^5*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt
(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(384*a^5*c^5 - (45*a*b^4*c^5 - 60*a^2*b^3*c^4*d +
346*a^3*b^2*c^3*d^2 - 340*a^4*b*c^2*d^3 + 105*a^5*c*d^4)*x^4 + 2*(15*a^2*b^3*c^5 + 109*a^3*b^2*c^4*d - 111*a^4
*b*c^3*d^2 + 35*a^5*c^2*d^3)*x^3 + 8*(93*a^3*b^2*c^5 + 22*a^4*b*c^4*d - 7*a^5*c^3*d^2)*x^2 + 48*(21*a^4*b*c^5
+ a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^5*x^5), 1/3840*(15*(3*b^5*c^5 - 5*a*b^4*c^4*d - 10*a^2*b^3
*c^3*d^2 + 30*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 7*a^5*d^5)*sqrt(-a*c)*x^5*arctan(1/2*(2*a*c + (b*c + a*d)*x)*
sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) - 2*(384*a^5*c^5 - (45
*a*b^4*c^5 - 60*a^2*b^3*c^4*d + 346*a^3*b^2*c^3*d^2 - 340*a^4*b*c^2*d^3 + 105*a^5*c*d^4)*x^4 + 2*(15*a^2*b^3*c
^5 + 109*a^3*b^2*c^4*d - 111*a^4*b*c^3*d^2 + 35*a^5*c^2*d^3)*x^3 + 8*(93*a^3*b^2*c^5 + 22*a^4*b*c^4*d - 7*a^5*
c^3*d^2)*x^2 + 48*(21*a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^5*x^5)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(1/2)/x^6,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.02, size = 967, normalized size = 3.42 \begin {gather*} -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (105 a^{5} d^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-375 a^{4} b c \,d^{4} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+450 a^{3} b^{2} c^{2} d^{3} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-150 a^{2} b^{3} c^{3} d^{2} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-75 a \,b^{4} c^{4} d \,x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+45 b^{5} c^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-210 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4} x^{4}+680 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3} x^{4}-692 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}+120 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d \,x^{4}-90 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4} x^{4}+140 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c \,d^{3} x^{3}-444 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{2} d^{2} x^{3}+436 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{3} d \,x^{3}+60 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{4} x^{3}-112 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{2} d^{2} x^{2}+352 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{3} d \,x^{2}+1488 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{4} x^{2}+96 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} c^{3} d x +2016 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,c^{4} x +768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, a^{2} c^{4} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(d*x+c)^(1/2)/x^6,x)

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^4*(105*a^5*d^5*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2))/x)-375*a^4*b*c*d^4*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
)/x)+450*a^3*b^2*c^2*d^3*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-150*a^2*b
^3*c^3*d^2*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-75*a*b^4*c^4*d*x^5*ln((
a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)+45*b^5*c^5*x^5*ln((a*d*x+b*c*x+2*a*c+2*(a*
c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/x)-210*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*d^4*x^4+680*(
a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c*d^3*x^4-692*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2
*b^2*c^2*d^2*x^4+120*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^3*d*x^4-90*(a*c)^(1/2)*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)*b^4*c^4*x^4+140*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c*d^3*x^3-444*(a*c)^(1/2)*(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c^2*d^2*x^3+436*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*c^3*d*
x^3+60*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^3*c^4*x^3-112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/
2)*a^4*c^2*d^2*x^2+352*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c^3*d*x^2+1488*(a*c)^(1/2)*(b*d*x^2+a
*d*x+b*c*x+a*c)^(1/2)*a^2*b^2*c^4*x^2+96*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^3*d*x+2016*(a*c)^(1
/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b*c^4*x+768*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(a*c)^(1/2)*a^4*c^4)/(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)/x^5/(a*c)^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(1/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(5/2)*(c + d*x)^(1/2))/x^6,x)

[Out]

int(((a + b*x)^(5/2)*(c + d*x)^(1/2))/x^6, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(d*x+c)**(1/2)/x**6,x)

[Out]

Timed out

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